Edges not following the mesh in edit mode?
You have a subdivision surface modifier on your mesh. By default that modifier does not change the display of the original mesh while in edit mode. Its personal preference I like to see the original mesh most of the time, but coming from max I can see why you want the edges on the mesh.

Just head to the Properties window, then the Modifiers tab and look a the  Subdivision modifier (Subsurf modifier in versions prior to 2.8).  Then simply tick the button highlighted in red "Adjust edit cage to modifier result."